We consider a model of fractional diffusion involving the natural nonlocal version of the $p$-Laplacian operator. We study the Dirichlet problem posed in a bounded domain $Ω$ of ${\mathbb{R}}^N$ with zero data outside of $Ω$, for which the existence and uniqueness of strong nonnegative solutions is proved, and a number of quantitative properties are established. A main objective is proving the existence of a special separate variable solution $U(x,t)=t^{-1/(p-2)}F(x)$, called the friendly giant, which produces a universal upper bound and explains the large-time behaviour of all nontrivial nonnegative solutions in a sharp way. Moreover, the spatial profile $F$ of this solution solves an interesting nonlocal elliptic problem. We also prove everywhere positivity of nonnegative solutions with any nontrivial data, a property that separates this equation from the standard $p$-Laplacian equation.

21 pages